Magic wavelength for a rovibrational transition in molecular hydrogen

Molecular hydrogen, among other simple calculable atomic and molecular systems, possesses a huge advantage of having a set of ultralong living rovibrational states that make it well suited for studying fundamental physics. Further experimental progress will require trapping cold H2 samples. However, due to the large energy of the first electronic excitation, the conventional approach to finding a magic wavelength does not work for H2. We find a rovibrational transition for which the AC Stark shift is largely compensated by the interplay between the isotropic and anisotropic components of polarizability. The residual AC Stark shift can be completely eliminated by tuning the trapping laser to a specific “magic wavelength” at which the weak quadrupole polarizability cancels the residual dipole polarizability.


S1. H 2 molecule in external, time-dependent electric field: perturbation theory approach
In this section, we derive the formulas for the correction to the energy of rovibrational levels in the ground electronic X 1 Σ + g state in H 2 due to an external, time-dependent electric field. We use time-dependent perturbation theory up to the second order with respect to the amplitude of the perturbing electric field, following the formalism outlined by Sambe 1 , and employ elements of the spherical tensor theory 2 . The derivation is based on the work of Gray and Lo 3 , who introduced spherical tensor formalism to the theory of static molecular polarizabilities of Buckingham and coworkers 4 and a review on rotational and vibrational contribution to molecular polarizabilities and hyperpolarizabilities written by Bishop 5 .
The Hamiltonian describing a molecule in an external, time-dependent electric field is a sum of the field-free molecular Hamiltonian,Ĥ 0 , and the perturbation,Ĥ ′ (t), which describes the interaction of molecular charges with external field H(t) =Ĥ 0 +Ĥ ′ (t). (1) The field-free molecular Hamiltonian yields the set of unperturbed eigenstates and energieŝ which will be specified later on. We consider the perturbation of molecular eigenstates due to the external, time-dependent, inhomogeneous electric field. The general form of the field is whereĒ is the mean electric field, related to the average intensity of the laser as I av = cε 0Ē 2 , ε is a complex polarization vector of the propagating laser beam, ν is the frequency of the laser field and k denotes the wavevector with magnitude k = |k| = 2πν/c. c and ε 0 denote the speed of light and vacuum permittivity, respectively.
The perturbation can be expressed as a Taylor series expansion with respect to a chosen position in the molecule 3,4 . Here, we restrict the expansion to the dipole and quadrupole termŝ where the Greek subscripts σ , τ = X,Y, Z denote the Cartesian components. We use uppercase X,Y, Z symbols to refer to the components defined with respect to the space-fixed frame of reference, and lowercase x, y, z symbols to describe the components defined with respect to the molecule-fixed frame. E σ and ∇E σ τ denote the components of the electric field and the electric field gradient at the origin of the expansion, respectively,μ is the electric dipole moment operator where the sum goes over all molecular charges q i and r iσ denotes the component of the position vector of the i-th charge.Q σ τ is the electric quadrupole moment operator, defined aŝ Here, we assume that the electric field is linearly polarized in the space-fixed Z-direction, ε = ε * =Ẑ, and that the wavevector is oriented towards the space-fixed Y direction, k = kŶ. In such a case, the perturbation is given aŝ It can be shown within the formalism of the time-dependent perturbation theory, that for the perturbation of the form there is no first-order correction to the eigenvalues of the unperturbed Hamiltonian,Ĥ 0 1, 6 . The second-order correction involves the cross products ofĤ ′ whereR (±) n (ν) is the frequency-dependent resolvent operator In the case considered here the second-order correction is given as where ⟨α ZZ (ν)⟩ n is the expectation value of the ZZ-component of the rank-2 dipole polarizability tensor, α σ τ (ν), in the |ψ and ⟨C Y Z,Y Z (ν)⟩ n denotes the expectation value of the Y Z,Y Z component of the rank-4 quadrupole polarizability tensor in the |ψ Note that we do not consider the contribution from the rank-3 dipole-quadrupole polarizability tensor since its expectation value vanishes for molecules with center of inversion, such as homonuclear diatomics 4,7 .
The goal of this derivation is the correction to the energy of rovibrational states of para-H 2 in the ground electronic X 1 Σ + g state. The eigenstates |ψ where |ηΛSΣ⟩ denotes the electronic state of the diatomic molecule, |ν(J)⟩ is vibrational state (we explicitly display its dependence on the rotational quantum number due to centrifugal distortion), and |JMΩ⟩ is the angular (rotational) part of the molecular state. S and L denote the total electronic spin and the total electronic angular momentum, respectively, and Σ and Λ are the projections of S and L on the molecular axis. The angular momentum associated with the rotation of the nuclei is denoted as R. R and L are coupled to form the intermediate angular momentum N. Since R is perpendicular to the internuclear axis, the projection of N on this axis is Λ. The total angular momentum (excluding nuclear spin, which is zero for para-H 2 ) is denoted as J = N + S = R + L + S, and its projection on the internuclear axis is Ω = Λ + Σ. In the absence of external fields the square of the total angular momentum and the space-fixed Z-component of the total angular momentum, commute with the Hamiltonian. The projection of J on the space-fixed axis is denoted as M. The space-fixed projections of R, L, N and S are denoted as M R , M L , M N , M S , respectively. The rotational part, |JMΩ⟩, is the Wigner D-function where ζ describes the set of Euler angles describing the orientation of the molecule-fixed reference frame relative to the space-fixed frame. For the ground electronic X 1 Σ + g state of H 2 , S = 0, Σ = 0, Λ = 0, and the notation is further simplified by putting J = N, M = M N and Ω = Λ = 0. The rovibrational levels of para-H 2 in the X 1 Σ + g electronic states are thus denoted as where the rotational part, Eq. (16), is simplified to a spherical harmonic denoted as |NM N ⟩, and |ν(N)⟩ is the solution of the radial rovibrational Schrödinger equation, given, in the coordinate representation as χ νN (r HH )/r HH , where r HH is the internuclear distance in H 2 . Substitution of Eq. (17) into Eqs. (12) and (13), leads to respectively. Note that as the electric dipole and quadrupole moment operators do not couple states with different S and Σ, these quantum numbers are omitted in the sum over primed states. As a consequence, the sum over Ω ′ = Λ ′ + Σ is also not written explicitly. In order to evaluate the matrix elements of the space-fixed components of the dipole and quadrupole moment operator in the basis introduced in Eq. (17), we expressμ Z andQ Y Z as p-labelled space-fixed components of irreducible spherical tensor operators of rank 1 and 2, respectively 8 We express the space-fixed elements of the spherical tensors a combination of the q-labelled molecule-frame components and the Wigner D-matrix We use a general formula for matrix elements of the D-matrix,

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to evaluate the matrix elements in Eqs. (19)-(20) Here, j 1 j 2 j m 1 m 2 m , is the Wigner 3-j symbol which vanishes unless m 1 + m 2 + m = 0. This implies that M ′ = M N + p and q = Ω ′ . The sums in Eqs. (19)-(20) are then split into two terms, which involve couplings with the excited electronic states, and the couplings with rovibrational levels in the S1.1 Dipole polarizability contribution to the Stark Shift ⟨α e ZZ (ν)⟩ X 1 Σ + g ,vNM N denotes the contribution to the dipole polarizability from couplings with excited electronic states where, due to symmetry properties of the 3-j symbols, Ω ′ , and thus Λ ′ , can take only three values: −1, 0, 1. Similar rules apply to J ′ , which equals N − 1, N and N + 1. Hence, if the laser frequency is far detuned from the electronic resonance, the denominator can be approximated as This leads to where and denote components of the dipole polarizability that are parallel and perpendicular to the molecular axis, respectively 9 . If we now introduce the isotropic and anisotropic parts of dipole polarizability as and

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respectively, we obtain the electronic contribution to the dipole polarizability in a well-known form .
One can similarly simplify the rovibrational contribution to the dipole polarizability, ⟨α rv ZZ (ν)⟩ X 1 Σ + g ,vNM N , which involves couplings with rovibrational levels in the Since H 2 is a homonuclear molecule, the transition dipole moment in a given electronic state vanishes, and there is no contribution to the dipole polarizability due to ⟨α rv ZZ (ν)⟩ X 1 Σ + g ,vNM N . The correction to the energy levels due to the dipole polarizability is thus given as .
In the present work we use the rovibrationally-averaged values of the isotropic and anisotropic parts of the dynamical dipole polarizability tensor reported in Ref. 10 . Note that the authors provided the values of ⟨α e (ν)⟩ X 1 Σ + g ,vN and ⟨γ e (ν)⟩ X 1 Σ + g ,vN for wavelengths in the range 182.25 − 1320.6 nm. For λ > 1320.6 nm we extrapolate the values from Ref. 10 using y(λ ) = a/(λ − λ 0 ) b + y DC , where y DC is the DC-limit value of ⟨α e (ν)⟩ X 1 Σ + g ,vN or ⟨γ e (ν)⟩ X 1 Σ + g ,vN reported in Ref. 10 , and a, b, λ 0 are the fitting coefficients. In the article we use a simplified notation for the dipole correction to the energy of rovibrational levels in the ground electronic state of H 2 with the term symbol of the electronic state made implicit for clarity.

S1.2 Quadrupole polarizability contribution to the Stark shift
In the case of the quadrupole polarizability, the contribution from the coupling with excited electronic states is given as The electronic contribution to the quadrupole polarizability in H 2 was studied in Ref. 11 . The authors provide the DC-limit values of the expectation values of the three independent components of the quadrupole polarizability tensor, defined in the moleculefixed frame of reference, in several rovibrational levels of H 2 . For instance, the DC-limit value of the ⟨C e zz,zz (ν)⟩ terms, is 6.518 e 2 a 4 0 E −1 h and 7.729 e 2 a 4 0 E −1 h for the |v = 0, N = 0⟩ and |v = 1, N = 2⟩ states in H 2 , respectively 11 . For the wavelengths in the vicinity of λ ≈ 2.413 µm we are far detuned from electronic resonances, which occur at extreme UV (see Fig. 1 for the dependence of electronic dipole polarizability on the frequency of the electric field). We can thus assume that the electronic contribution to the dynamical quadrupole polarizability can be approximated by its DC counterpart. Substituting k = 2π/λ to Eq. (11), we obtain the difference between the correction to the energy of the |v = 0, N = 0⟩ and |v = 1, N = 2⟩ states in H 2 due to the electronic part of the quadrupole polarizability to be at the level of 10 −10 e 2 a 2 0 E −1 h , and we neglect this contribution in the further analysis.

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The correction to the energy levels due to the quadrupole polarizability is thus given only by the coupling with rovibrational levels of the X 1 Σ + g state where The sum over N ′ is restricted by the 3-j symbol to only three terms: N ′ = N + 2 and N ′ = N − 2, and N ′ = N, which correspond to the S, O and Q branches, respectively. The latter term vanishes if v ′ = v (the pure rotational Q lines do not exist). Eq. (41) can be compared with Eq. (224) in Ref. 5 , which presents the rovibrational contribution to the static quadrupole polarizability. Note that the authors consider the C rv ZZ,ZZ component, for which the only non-vanishing element of the sum over M ′ involves the M ′ = M N term.
Here, we calculate the rovibrational contribution to the quadrupole field gradient polarizability using rovibrationallyaveraged quadrupole transition moments reported in Ref. 12 and the transition frequencies for rovibrational transitions in H 2 obtained from the H2Spectre code of Czachorowski et al 13 and Komasa et al. 14 . In the article we use a simplified notation for the correction to the energy due to the quadrupole polarizability we drop the term symbol for the ground electronic state whenever possible, we introduce the frequency of each rovibrational resonance, hν v ′ N ′ ←vN = E X 1 Σ + g v ′ N ′ − E X 1 Σ + g vN , and the quadrupole transition moment function Q(r HH ) 12 If the frequency of the electric field is close to the resonant frequency ν v ′ N ′ ←vN , the rovibrational contribution in Eq. (41) can be approximated with a single term, and the correction to the quadrupole polarizability can be written as is the detuning from the resonance, and the Rabi frequency is defined as

S2. Selection rules for quadrupole transitions
The strength of the 1-0 S(0) electric quadrupole transition is related to the matrix element of the quadrupole term in the Hamiltonian that couples the initial and final spectroscopic states. Using irreducible spherical tensor operators, we can write an expand the scalar product in the space-fixed frame of reference The space-fixed p-labelled components of the quadrupole moment are transformed to the molecule related to the q-labelled molecule-fixed components through the Wigner D-matrix. Using Eq. (23) and the simplified notation introduced in Eq. (43), we obtain The 3-j symbol indicates that the desired ∆M N = ±2 transition is driven by the p = ±2 components of the electric field gradient tensor. Moreover, the remaining two components of the 1-0 S(0) transition, ∆M N = ±1 and ∆M N = 0, can be eliminated if the electric field of the probe laser is oriented in such a way, that the p = 0 and p = ±1 components of the electric field gradient tensor vanish. The relation between spherical and Cartesian components of the field gradient, T p (∇E) and ∂ α E β , is given as 3 For the electric field of the form the Cartesian elements of the electric field gradient tensor are simply Thus, if the direction of propagation of the probe beam is set to the space-fixed Y axis, k = kŶ, and the probe beam is linearly polarized in the space-fixed X direction, the electric field vector is given as the field gradient has only one non-zero Cartesian component and two desired non-zero spherical components

non-perturbative approach based on the denisity matrix equation for a three-level system
The perturbation approach to the analysis of the rovibrational structure of the H 2 molecule in the external time-dependent electric field is limited to the frequency ranges that are significantly detuned from electronic or rovibrational resonances, ∆ ≫ Ω.
The magic wavelength resulting from the cancellation of the residual dipole polarizability with the quadrupole polarizability is found for field frequencies detuned by less than 0.5 MHz from rovibrational resonances. Thus, we study the shape of the 1-0 S(0) resonance in the strong, external electric field in a non-perturbative way, based on the density matrix equation for the three-level system in a Λ-configuration, presented in Fig. 1. Level |v = 0, N = 0, M N = 0⟩ ≡ |1⟩ is coupled with |v = 1, N = 2, M N = ±2⟩ ≡ |3⟩, level |3⟩ is coupled with level |v = 0, N = 2, M N ⟩ ≡ |2⟩, and there is no coupling between levels |1⟩ and |2⟩.
The Hamiltonian for the system is a sum of the field-free part,Ĥ 0 , and the perturbation of the form introduced in Eq. (7), which involves the contributions from the two perturbing fields. The field that couples the |1⟩ and |3⟩ states and drives the 1-0 S(0) transition is linearly polarized in the X direction with and the field that couples the |2⟩ and |3⟩ states (the trapping field) is linearly polarized in the Z direction with k t = k tŶ Note that in this Section we use angular frequencies, ω = 2πν, which are more convenient for solving the density matrix equations. The analysis in this Section is restricted to only three levels in the ground electronic state of H 2 -the dipole term in Eq. (7) is zero, and the three levels are coupled by non-diagonal elements ofĤ ′ (t), which originate from the interaction of the molecular quadrupole moment with the electric field gradient. The Hamiltonian is thus given aŝ wherehω 31 = E 3 − E 1 andhω 32 = E 3 − E 2 are the (positive) frequencies of the 1-0 S(0) and 1-0 Q(2) resonances, and the Rabi frequencies are defined as with the quadrupole moment operator,Q, defined in Eq. (6). The definition of the Rabi frequency for the trapping field is equivalent to Eq. (45). The frequencies of the lasers are assumed to be close to the corresponding resonances, hence, the we use the detunings from the unperturbed transition frequencies

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The evolution of the three-level system is governed by the simplified Liouville equatioṅ whereΓ is a phenomenological operator describing the relaxation from the excited states and the damping of the coherences.

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where n 31 = ρ 33 − ρ 11 , and n 32 = ρ 33 − ρ 22 are the population differences between the levels, and where is introduced for the sake of brevity. The steady-state populations of the levels are found from the first three equations in (62), puttingρ kk = 0. Using additionally the condition and the fact that the Rabi frequencies, as defined in Eq. (58), are purely imaginary, we obtain The real parts of the coherences from (67) are given as Reρ 32 = n 32 |Ω t |Γ where we introduced Substituting (71) to (70) leads to the final formula for the difference between the populations of the state |1⟩ and |3⟩ and for the difference between the populations of the state |2⟩ and |3⟩ n 32 = n 31 where we have introduced .
Note that such form of the formula for n 31 is convenient if one wants to recover the population difference for the two-level system 16 , by puttingΩ t = 0 The explicit formulas for the coherences are found by substituting Eqs. (73) and (74) to (67). Of particular importance is the real part of the steady-state SVEA coherence between states |1⟩ and |2⟩, which determines the shape of the 1-0 S(0) transition induced by the probe laser. Substituting Eq. (74) to (71) leads to Reρ 31 = n 31 |Ω p |Γ

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Similarly to Eq. (73), this particular form is convenient if one wants to recover the result for the two-level system Reρ 31 = n 31 |Ω p |Γ which describes the power-broadened Lorentzian shape of an optical resonance.
Here, we determine the shape of the 1-0 S(0) resonance from Eq. (77) for various detunings of the trapping beam, ∆ t , assuming that the Rabi frequency of the probe beam is significantly smaller than the Rabi frequency of the trapping beam. The value of Γ is fixed at 0.1 MHz.